SASci Seed Session, Santa Fe

Page 1292 of von Foerster et al. (1960, Science) is reproduced below to show the log-log fit (power law, foursquare) of world population 1750-1960, with population on the Y (left) axis such that log-growth is a linear function of log population density (Y right axis, right) as related by a time (X axis)-dependent formula, with data points fitted by least-squares to the regression line. The equation that relates population growth dN/dt to density (roughly linear with N squared) has an instability (blowup region) before reaching the year 2027, hence the model predicts a global demographic transition and also locally (at different time period, because of spatially heterogeneous densities), before the point of global instability is surpassed. Because of the log-log relations, this is a scale-free model (up to blowup) of a time-driven power-law and thus a scalable process which, in principle, could apply to earlier time periods as well. Linear relationships in log-log plots enable goodness of fit of empirical data points with ordinary least squares. Some of the population macro-models in this session will use this or related approaches. Plotting of country specific population densities (Y axis, right) is loosely related to the idea that different countries will go through the demographic transition at different times, but that aspect of the theory is not elaborated in this model, and would require additional parameters. Instead, this model is aimed at the basic scale-independent aspect of human population growth, within finite limits. This type of dynamical model (dynamic not because it is longitudinal, which is not sufficient to constitute a dynamic, but because it predicts instabilities as a function of time for a process that is uniform up to instability) is common in physics. Lest we think that physics has no relation to anthropology, recall that von Foerster was President of the Wenner Gren Foundation of Anthropological Research from 1964 to (1968?). Note that humans are the only species with power-law population growth. Population growth in all other species is observed to follow a simple exponential with a characteristic scale, and populations often cycle around that scale, as with predator/prey and Malthusian cycles. If we look closely at the von Foerster graph we will also see a population cycle (in fact, 1.5 cycles) in the period 1700-1960.

One of the explanations given for the unusual power-law growth of the human species given by von Foerster is the higher connectivities among humans in terms of information flow, enabling technological developments through invention and diffusion, and also enabling humans to form lesser or grand collaborative coalitions in a game against nature to improve (or destroy) the carrying capacity of the environment. Von Foerster treats this connectivity as a relative constant in the time period of this graph, but comments that this may be a further variable to examine in connection with such models at earlier time periods.

Part of the reason for cycling around a von Foerster regression line in human population growth may be temporary Malthusian constraints of reaching environmental limits. Another is sociopolitical violence, which seems to kick in as these constraints are reached, among other factors.

Part of the objective of the macromodeling papers in this symposium should be to replicate the von Foerster graph with current data that also reachs further back in time to include historical and archaeological time periods, and to investigate whether there are deviations from this model, either at transition points (phase transitions) or in cyclical deviations, amplification or reduction of cyclical deviations, and lengthening or shortening of cycles. Future research will need to test whether this type of model replicates regionally, and test the effects of changes in connectivity. The degree of multiconnectivity in trade and/or warfare networks, for example, may prove to be an important variable. Click the image to enlarge and print.
Kraden paper
Thurs Feb 24th - General Program
Seed Session
Foerster.pdf 1960
Doomsday 1961
Pop Den & Growth 1961
DornOnFoerster.pdf 1962
Foerster1962.pdf 1962
Von Foerster at Wenner Gren 1964
biography of Von Foerster 1964
SerrinDoomsday.pdf 1975
Umpleby on Foerster 2001
Foerster Bibliography 2001

Nathan Keyfitz. 1975. How Do We Know the Facts of Demography?, Population and Development Review 1(2): 267-288. Population Growth and Earth's Human Carrying Capacity

Joel E. Cohen 1995. Science 269(5222): 341-346.

Kapitza on Quadratic Growth

Taagepera article abstract Conference 2004 abstract

From Fri Dec 17 11:31:10 2004
Date: Fri, 17 Dec 2004 09:28:25 -0800 (PST)
From: Doug 
To: Andrey Korotayev 
Cc: Artemy Malkov , Daria Khaltourina ,
     Douglas R White 
Subject: Re:

the pdfs can each be downloaded from the web. They include letters and
reponses by Foester et a, later commentary. Foester covers almost all of
what we have talked about (including technology, dynamical instabilities
and transition points, ect.). Totally consistent with Artemy's framework.
Here is my current thinking.

All but two of Foerster's data points are between 1750 - 1960.
As with Artemy's framework, Foerster indicates that where N --> infinity
there is an instability point.

The data now compiled goes back to -10,000 and further.
For the period prior to 500BCE we can ask and computed whether there is
the same type of log-log curve for population and population density.
If we plot our data on his p.1294 curve that seems a good way to begin.
The data point for 500BCE is very close to his line. But as we go back to
8000BCE our points go down very steeply to fit his curve. One hypothesis
is that our data are inaccurate, underestimates as we go further back in
time. Another is that Foester's equation sets are correct, but there is a
transition point where the constants in the equations change.

Do we also have the data on areas for our early data fit to his curve,
i.e., calculating population density?

Say we concluded that Foester's equation sets (3)-(9) hold, but with
different constants (10)-(13). Then what accounts for the discontinuity? One
possibility is that there was an instability near 500BCE with these
constants. But is that date anywhere near an instability for N -->

Other are that connectivity changed. Or technology changed. But once we go
this route, since these are also gradual changes, we need to account for
criticalities in one or more of the variables we introduce. And these
should relate to the "grand coalition" idea of Foester perhaps, or game
against nature.

Because the curves before and after 500BCE are so great, I think these are
the questions we need to address first, along with the question of how
accurate are the estimates for the early period.

If we are successful for this transtion, then we can apply the same
approach and questions to the period from 500BCE to approximately 1500 CE
plus or minus 200 years, where there is another more subtle transition.

I would not like to hear more talk about changes in world system,
expressed quantitatively. I would like to see all such arguments phrased
in terms of measureable variables.

Doug White, professor, UC Irvine

From Fri Dec 17 08:38:43 2004
Date: Mon, 13 Dec 2004 06:23:36 +0300
From: fabr 
To: Douglas R. White 
Cc: KOROTAYEV@YAHOO.COM, Andrey Korotayev ,
     Peter Turchin , Don Saari 
Subject: Re[2]: some loose ends

Let me get into your discussion. It seems to me that my physical point
of view will assist you in clarification of the macromodel

Dynamics of every physical body is influences by a huge number of
factors. Modern physics abundantly evidences it. Even if consider a
falling ball, we inevitably face such forces as gravitation, friction,
electromagnetic forces, forces caused by pressure, by radiation, by
anisotropy of medium and so on.

All these forces do have effect on motion of considered body. It is a
physical fact. Consequently in order to describe this motion we should
construct an equation involving all these factors. Only in this case
we may "guarantee" the right description. Moreover, even these
equation is not right! Because we have not included the factors and

It is evident that such a puristic approach and a rush for precision
lead to agnosticism and nothing else.

Fortunately, from the physical point of view, all the processes have
their characteristic time scales and their application conditions.

Even if there is a great number of significant factors we can
sometimes neglect all of them except the most evident one.

There are two main cases of simplification:
1. When force, caused by selected factor is much more than all other
2. When selected factor has a characteristic time scale which is
   adequate to the scale of considered process, while all other
   factors have much different time scales (less or more - does not

The first case seems to be clear.
As for second, it is substantiated by Tikhonov theorem (1952).
It states that if there is a system of three differential equations,
and if the first variable is changing very quickly, the second changes
very slowly, and the third is changing with an acceptable characteristic
time scale, then we can discard the first and the second equations and
pay attention only to the third one. In this case first equation must
be solved as a algebraic equation (not as a differential), and the
second variable must be handled as a parameter.

Let's consider some extremely complicated process.
For example, photosynthesis.
Characteristic time scales are the following (in seconds):

1. light absorption:              ~ 0.000000000000001
2. reaction of charge separation: ~ 0.000000000001
3. electron transport:            ~ 0.0000000001
4. carbon fixation:               ~ 1 - 10
5. transport of nutrients:        ~ 100 - 1000
6. plant growth                   ~ 10000 - 100000

Such a spread in scales allows constructing rather simple and
valid models for every scale without involving all these processes
into consideration.

Each time scale has its own laws and equations that are limited by
the corresponding conditions.
If the system exceeds the limits of respective scale, its behaviour
will change, and equations will also change. It is not a defect of
description - it is just a transition from one regime to another.

For example, solid bodies can be described perfectly by solid
models having respective equations and laws of motion (e.g. mechanics
of rigid body) , but increasing the temperature will cause melting process,
and the same body will convert into liquid, which must me described by
absolutely different laws (e.g. hydrodynamics), finally the same body
converts into gas and obeys respective laws (e.g. Boyle's law etc.)

It may look like a mystification. The same body may obey different laws
and equations when temperature changes SLIGHTLY!
But it is a fact.
Moreover, from the microscopic point of view - all these laws
originate from micro interaction of molecules, which remain the same
for solid body, liquids and gases.
But from the point of view of macro processes, macro-behavior is
different and equations are different.

So there is noting abnormal that the dynamics of complex system is
described has phase transition and sudden changes of regimes.

For every change in physics there are always limitations that change
the law of change in the neighborhood of limit. Examples of such
limitations are absolute zero of temperature and velocity of light.
If temperature is big enough or, respectively, velocity is small, then
classical laws are working perfectly, but if temperature is close to
absolute zero or velocity is close to velocity of light - then behavior
may change incredibly. Such effects as superconductivity or space-time
distortion may be observed.

As for demographic growth, there are a number of limitations, each of
them having their characteristic scales and applicability conditions.

Analyzing the system we can define some of these limitations.
Growth is limited by:

1. RESOURCE limitations:

1.1. starvation - if there is no food (or other significant for vital
     functions resources) there can be no growth, but collapse
      time scale ~ 0.1 - 1 year
      conditions: RESOURCE SHORTAGE

This is a strong limitation and it works inevitably.

1.2. technological - technology may support a limited number of workers
       time scale of ~ 10-100 YEARS

this is a relatively rapid process, which causes demographic cycles.


2.1. birth rate - a woman cannot produce more than 1 child per year
     (while a man can do :)
     time scale ~ 1 year

This is a very strong limitation with a short small time scale,
so it will be the only rule of growth if for any of possible reasons
respective condition (birth rate is very high) will be observed.
2.2. pubescence - a woman cannot produce children until she is mature
     time scale ~ 15-20 years
     conditions: EARLY CHILD-BEARING

This condition is less strong than 2.1. but in fact condition 2.1.
is rarely observed. For real demographic processes limitation 2.2. is
more essential than 2.1. because for premodern societies women start
giving birth being 15-20 years old.


3.1. infant mortality - mortality is obviously preventing population growth
     time scale ~ 1-5 years

Short time scale - strong and actual limitation for premodern societies.

3.2. mobility - in preagrarian societies woman cannot have many
         children, cause it reduces mobility.

     time scale: ~3 years
     condition: WANDERING TRIBES

3.3. education - education increases the "cost" of individual, but
     requires years of education, making the procreation undesirable.
     High human-cost allows an educated person to stand on his own
     legs until his old age, without help of his children. These
     limitations reduce birth rate.

     time scale: ~25-40 years
     condition: HIGH EDUCATION

All these limitations are objective. But each of them is ACTUAL (that
is it must be included into equations) ONLY IF RESPECTIVE CONDITIONS

If for any considered historical period several limitations are
actual (under their conditions) then, neglecting the others,
equation for this period must involve their implementation.

According to Tikhonov theorem the most strong of them are ones having
shortest time scale.
HOWEVER, factors with longer time scale may "start working" under
less severe requirements, making short-time-scale factors not actual,

Lets observe and analyze the following epochs:

I.   pre-agrarian societies
II.  agrarian societies
III. post-agrarian societies

using such a notation:

- atypical - means that the properties of the epoch
  make the conditions practically impossible.
- actual - means that such conditions are observed, so
  this limitation is actual and must be involved into implementation

- potential - means that such conditions are not observes
  but if some other limitations will be removed, this limitation
  may become actual

I. pre-agrarian societies
1.1. - ACTUAL *
1.2. - ACTUAL **
2.1. -   potential
2.2. - ACTUAL
3.1. - ACTUAL
3.2. - ACTUAL
3.3. -   atypical

II. agrarian societies
1.1. - ACTUAL
1.2. - ACTUAL
2.1. -   potential
2.2. - ACTUAL
3.1. - ACTUAL
3.2. -   atypical
3.3. -   potential

II. post-agrarian societies
1.1. -   atypical
1.2. - potential/actual ***
2.1. -   potential
2.2. -   potential
3.1. -   atypical
3.2. -   atypical
3.3. - ACTUAL

* systematic (not an occasional short term) starvation is caused
       by misbalance of technology and population, so 1.1. may be
       included into 1.2.

** according to Tikhonov theorem we may neglect the oscillations
       of population (demographic cycles), because their time scales
       are at least 10 times less than the scale of historical period
       that is taken into account.
*** technology produces much more than it is necessary for the sustenance,
       but the living standards also require more resources

In our macromodel we involve only agrarian and post-agrarian societies
(for leak of data for preagrarian societies)

According to Tikhonov theorem to describe the DYNAMICS of the system
we should take actual factor, which has the LONGEST time-scale (it will
represent dynamics, while shorter scale factors will be involved as
coefficients - solutions of algebraic equations),

So epoch [II] is characterized by 1.2, and [III] - by 3.3
([III] also involves 1.2, but for [III] resource limitation
1.2 is much less essential, cause it concerns life standards,
and not vitally important needs)

Thus, demographic transition is a process
of transition from II:[1.2]  -->  to III:[3.3]

Limitation 3.3 at [III] makes biological limitations unessential but
potential. (possibly, in future, limitation 3.3 could be reduces, for
example, reducing time of education thanks to higher educational
technologies, so it will make [2.2] actual again; possibly cloning
will make [2.1] and [2.2] obsolete - so there will become apparent
new limitations)

In conclusion, I want to note, that hyperbolic growth is a feature
which corresponds to II:[1.2], there is no contradiction
between hyperbolic growth itself and [2.1] or [2.2]. Hyperbolic agrarian
growth does never reach the birth-rate, which is close to conditions
of [2.1]. If it was so, hyperbola will obviously convert into
exponent, when birth-rate will come close to [2.1] (just as physical velocity
may never exceed the velocity of light) - and it would be not a
weakness of model, just a common sense. It would be [1.2] -> [2.1, 2.2]

But actual demographic transition [1.2] -> [2.1] is more drastic than
this [1.2] -> [2.1, 2.2]!

[3.3] is reducing birth-rate much more actively, and it may seem strange
SLOWER - during the epoch of [II]!

(it is not nonsense because slower growth was the reason of [2.1]
and [3.1])

As for the after-doomsday dynamics, - I said - if there is no
resource of spatial limitation, (as well as [3.1]) - then the [2.1]
and [2.2] will become actual. If they will also be removed (cloning,
etc.) then there will appear new limitations.
But if we consider the solution of C/(t0 - t) just formally - the
after-doomsday dynamics has no sense. But it normal, such as
temperature below absolute zero, or velocity above velocity of light
have no sense.

As for statistical problem of "one series" - it actually exists!

however, the stability of the system (perfect hyperbola for
population and pretty-good GDP hyperbola) gives us hopes for macro-law
definition. The more serious problem having one series is to
distinguish which is primary - GDP or population and which is
I am absolutely agree, that the most important direction of future
research is to find and analyze "several independent series"
(dynamics for pre-Columbian Afroeurasia, America, Australia etc.)
even through it will be an arduous task.


DRW> Right - but I disagree in the following way. Consider removing the
DRW> constraints the way that Newton does for the falling apple to get the
DRW> gravity results. That is, would the hyperbolic, now in discrete form with
DRW> ca. 20 year generations on average between births still make sense IF
DRW> extrapolated to AFTER DOOMSDAY? Would world population INTRINSICALLY begin
DRW> to fall semi-hyperbolically and in reverse? Absolutely not! This is what
DRW> you have to do when you look for "mechanisms" underlying equations.

DRW> Say you fit the equation for an orbit around the sun. Now take away the
DRW> sun. Do you suppose the movement will follow the orbit just because you
DRW> have an equation? I.e., don't privilege the equation over the physics and
DRW> the experimental method of the sciences generally.

DRW> As Saari and Turchin would say, you have to lay out a large potential set
DRW> of possible dynamic equations delta Y = f(A,B,C) where you think you know
DRW> the things -- MEASURABLE THINGS -- that affect changes in Y, but there are
DRW> a variety of ways that these may interrelate with an unknown number of
DRW> stable and unstable equilibria. The problem is NOT JUST TO ASSUME ONE
DRW> SERIES (your fallacious macromodeling approach) but to investigate using
DRW> measurable variables multiple cases where you can observe the dynamics of
DRW> interactions among these observables.

>> Dear Doug,

DRW> The correct interpretation of von Foerster's finding would have been just
DRW> that the world demographic growth reached such a point that its basic
DRW> pattern was bound to get radically changed well before 2026. And this
DRW> prediction would have started to get confirmed within a very few years
DRW> after 1960. We have rather frequently growth patterns described accurately
DRW> by certain equations, which when extrapolated produce absurd results, and
DRW> this just says that the respective growth pattern will get changed well
DRW> before the date of predicted absurd result. Take for example the simplest
DRW> demographic model dN/dt = aN. Does it make sense? Yes, it does, as,
DRW> ceterum paribus, 10 million mothers would give birth to approximately 10
DRW> times more babies than 1 million mothers. Do we observe things like 2%
DRW> annual growth rates for some periods of time? However, what will we have
DRW> if we extrapolate this pattern observed in a certain country (of say 10
DRW> mln people) in future? Just in 350 years we will have 10 bln people, in
DRW> 700 years 10 trillion, and in 1050 we will have 10000 trillion. Is this
DRW> result absurd? Perhaps, somehow. But in fact, it just shows that the
DRW> respective growth pattern cannot last for long, as, e.g., "ceterum
DRW> paribus" implies the same level of resource provision and the country
DRW> resource growth 1 billion times is simply impossible (let alone lots of
DRW> other things). This simply means that the respective growth pattern will
DRW> get transformed in a fairly near future. The same goes for hyperbolic
DRW> growth patterns (both in physics and social life), which are bound to get
DRW> transformed into non-hyperbolic dynamics at certain points.